Form the differential equation of the family of parabolas having a vertex at the origin and axis along positive $y-axis.$

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Form the differential equation of the family of parabolas having a vertex at the origin and axis along positive $y-axis.$

Equation of the family of parabolas having a vertex at the origin and axis along positive $y$ -axis can be represented by
$(x)^{2}=4 a y$ , where $a$ is an arbitrary constants.
$x^{2}=4 a y(1)$
Differentiating the above equation with respect to $x$ on both sides, we have,
$\begin{array}{l} 2(\mathrm{x})=4(\mathrm{a}) \frac{\mathrm{dy}}{\mathrm{dx}} \\ \mathrm{x}=2 \mathrm{a} \frac{\mathrm{dy}}{\mathrm{dx}} \\ \mathrm{a}=\frac{\mathrm{x}}{2 \frac{\mathrm{dy}}{\mathrm{dx}}} \end{array}$
Substituting the value of $a$ in equation (1)
$\begin{array}{c} \mathrm{x}^{2}=4 \frac{\mathrm{x}}{2 \frac{\mathrm{dy}}{\mathrm{dx}}} \\ \mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}=2 \mathrm{y} \end{array}$
This is the required differential equation.